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distance

Source: Taiwan MO 1998, Problem 4

August 26, 2012
geometryincentergeometry unsolvedTaiwan1998#4geometric inequality

Problem Statement

Let II be the incenter of triangle ABCABC. Lines AIAI, BIBI, CICI meet the sides of △ABC\triangle ABC at DD, EE, FF respectively. Let XX, YY, ZZ be arbitrary points on segments EFEF, FDFD, DEDE, respectively. Prove that d(X,AB)+d(Y,BC)+d(Z,CA)≤XY+YZ+ZXd(X, AB) + d(Y, BC) + d(Z, CA) \leq XY + YZ + ZX, where d(X,ℓ)d(X, \ell) denotes the distance from a point XX to a line ℓ\ell.
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